Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

Abstract

<p style='text-indent:20px;'>We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.</p>